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Pۘ_  _(     0xup0* MProbability Distribution  2(   6yjJ"`p~  The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number x by p(x) = S  2$$$$.$$$$$$$$$,;;`   c $A ?? P0  H  0޽h ? 3380___PPT10.`kf $f(  $ $ 0苗P 0  ['Parameter of a Probability Distribution( 2((k $ 6HjJ"`  SSuppose that p(x) depends on a quantity that can be assigned any one of a number of possible values, each with different value determining a different probability distribution. Such a quantity is called a parameter of the distribution. The collection of all distributions for all different parameters is called a family of distributions.T 2 $$$$$ $d$$$H $ 0޽h ? 3380___PPT10.0f ~((  ( ( 0̟P@  T Cumulative Distribution Function! 2!( ( 0, uyThe cumulative distribution function (cdf) F(x) of a discrete rv variable X with pmf p(x) is defined for every number by z 2+$$$$$$ $$$$"$>&&` ( c $A ??pP@=  o ( 0䰗@  YFor any number x, F(x) is the probability that the observed value of X will be at most x.Z 2$$$$$$0$$$$$f ( 0jJ"`PH ( 0޽h ? 3380___PPT10.4   ,(  , , 0P  ? Proposition 2 ( , 0Pŗ !For any two numbers a and b with L" 2$$$$$` , c $A ??= ` , c $A ??0@ x , 0˗   a  represents the largest possible X value that is strictly less than a. fL 2$$#$$"$$$  , 0Tӗ d  GNote: For integers 2$`  , c $A ??P @7 f  , 0jJ"`@ H , 0޽h ? 3380___PPT10.]c0 ((X(    0,ܗP0> z2Probability Distribution for the Random Variable X,30 21$$  0p' 3A probability distribution for a random variable X:840 21n @4  # #"U@4   <T?0@4  N0.09 @`  <L?04  N0.11 @`  <8? 4  N0.15 @`  <?p  4  N0.20 @`   <?0 p 4  N0.17 @`   <p?0 4  N0.15 @`   <?4  N0.13 @`    <?4  P(X = x)\  @`   <D1?0@ K6 @`  <8?0 K4 @`  <d@?  K1 @`  <tH?p   K0 @`  <P?0 p  N 1 @`  <K?0  N 3 @`  <tZ? P 8  @`  <g? Ox @``B  0o ?@ZB  s *1 ?@`B  0o ?4 @4 `B  0o ?4 ZB  s *1 ?4 ZB  s *1 ?4 ZB  s *1 ?0 0 4 ZB  s *1 ?p p 4 ZB  s *1 ?  4 ZB  s *1 ?4 ZB  s *1 ?004 `B   0o ?@@4  ! 0hlP   >Find0 2` " c $A. ??    # 6p>"`  ,$D 0 80.65 2B $ 0D>` ` ,$D  02 % 0>"` ` ,$D  0 & 6`t>"`P ,$D 0 80.67 2B ' 0D> p P ,$D  02 ( 0>"` ,$D  0H  0޽h ? ̙33f^___PPT10>+<D' = @B Dm' = @BA?%,( < +O%,( < +DN ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-g6B fade*<3<*#D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*#D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bwipe(down)*<3<*$D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(down)*<3<*%DN ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*&D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-s6Bwipe(down)*<3<*'D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(down)*<3<*(+p+0+# ++0+& + p(  p p 6l8c"`0p'  s13.3 Expected Values of Discrete Random Variables &2(20.<H p 0޽h ? 3380___PPT10. Pۘ<  0<(  0 0 0P   eThe Expected Value of X2 2((( 0 0 @p  sLet X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted t 2$$.$$ $$$$$$$ $$$ $,#8` 0 c $A  ??P    ` 0 c $A  ??@0   f  0 0jJ"`H 0 0޽h ? 3380___PPT10.0s*J7' {s$$ (    0 p sEx. Use the data below to find out the expected number of the number of credit cards that a student will possess. .t0 2q @ # #"2& *@  <?t@ N0.01 @`  <|ß?U @t N0.03 @`  <@џ?6 @U  N0.06 @`  <,ӟ? @6  N0.16 @`  <۟? @  N0.38 @`   <$?@  N0.28 @`   <4?@ N0.08 @`    <?@ P(x =X)\ @`   < ?t K6 @`   <l?U t K5 @`  <?6 U  K4 @`  <? 6  K3 @`  < %?   K2 @`  <,?  K1 @`  <P/? K0 @`  <x=? Ox @`ZB  s *1 ?@ZB  s *1 ?@ZB  s *1 ?@ZB  s *1 ? @ ZB  s *1 ? @ ZB  s *1 ?6 @6 ZB  s *1 ?U @U ZB  s *1 ?t@tZB  s *o ?@ZB  s *o ?ZB  s *1 ?ZB  s *o ?@@  0AW Zx = # credit cards,0 2 ! c $A0 ?? '8 $D  0 " c $A1 ??p  8 $D  0 # 0pFp ,$  0 ?=1.970 2 $ 6I>P @ ,$D 0 NAbout 2 credit cards0 2H  0޽h ? ̙33d\___PPT10<+D' = @B Dk' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-s6Bwipe(left)*<3<*!D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(left)*<3<*"D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$+p+0+#m ++0+$m +% 4%(  4 4 0] p T The Expected Value of a Function! 2!( 4 0k @p  qIf the rv X has the set of possible values D and pmf p(x), then the expected value of any function h(x), denotedr 2 $$ $$ $$$$ $$$$$$ $,(>` 4 c $A ??P @  ` 4 c $A ??@0  f 4 0jJ"`H 4 0޽h ? 3380___PPT10.0s*J F> 8(  8 8 0@xP0P  ORules of the Expected Value 2(` 8 c $A ??S  8 0|`` PH@___PPT9" MThis leads to the following: For any constant a, 2. For any constant b, x 2 2 2$$$$$$8X` 8 c $A "?? `<  "` 8 c $A %??@`  %H 8 0޽h ? 3380___PPT10.pEx. The quiz scores for a particular student are given below:.?0 2<  00pw 222, 25, 20, 18, 12, 20, 24, 20, 20, 25, 24, 25, 1830 23 @`  0\p c)Find the variance and standard deviation.*0 2*  c $A@ ??P @ 8 $D  0 @0  # #"0@ ,$D 0  < ?v I3 @`  <̣?v I2 @`   <ӣ? v I1 @`   <$ۣ?@ v  I4 @`   <d?# v@  I2 @`   <? v#  I1 @`   <?@v  S Frequency   @`  < ?  K.23 @`  <x?  K.15 @`  < ?   K.08 @`  < ?@    K.31 @`  <?# @   K.15 @`  <? #   K.08 @`  <4 ?@   U Probability   @`  <.?0v J25 @`  <5?0v J24 @`  <P7? 0v J22 @`  <PD?@ 0 v J20 @`  <0L?# 0@ v J18 @`  <\M? 0# v J12 @`  <T[?@0 v PValue  @``B  0o ?@00ZB  s *1 ?@vv`B  0o ?@  `B  0o ?@0@ ZB   s *1 ? 0  ZB ! s *1 ?# 0#  ZB " s *1 ?@ 0@  ZB # s *1 ? 0  ZB $ s *1 ?0 ZB % s *1 ?0 `B & 0o ?0 ZB ' s *1 ?@ ( s *pP,$D  0 ) c $AA ??p @G1 8 $@  0 * c $AB ?? 8 $D  0H  0޽h ? ̙33'___PPT10+o_pD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*)D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<**D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<**D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bwipe(left)*<3<*(+9 P(    c $AC ??Fr8 $D 0  0AD ?3f>?88 $D  0  c $AE ??P,8 $D 0  0AF ?3f>?P 8 $D  0H  0޽h ? ̙33___PPT10+:ֹ`D' = @B DP' = @BA?%,( < +O%,( < +D3' =%(%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*+ =5@(  @ @ 6jJ" 0: QShortcut Formula for Variance 2(` @ c $A ,??0p| ,` @ c $A 1???  1H @ 0޽h ? 3380___PPT10.Px ` Dx(  D D 6jJ" P0  ERules of Variance 2(` D c $A 4??`q 4` D c $A 7??  7 D 6LjJ" d 7and 2$ D 6jJ" p  PThis leads to the following: 2$`  D c $A 8??@} 8`  D c $A >?? 0sJ  >`  D c $A A?? 0  AH D 0޽h ? 3380___PPT10.P=. pt.(  t t 68c"`p  23.4 The Binomial Probability Distribution 43(20.<<H t 0޽h ? 3380___PPT10. PۘF HF(  H H 6jJ" p@ GBinomial Experiment 2(* H 6 jJ"  dAn experiment for which the following four conditions are satisfied is called a binomial experiment.6e 2P$$$ H 60jJ" _D<4___PPT9 e The experiment consists of a sequence of n trials, where n is fixed in advance of the experiment. Nf 2,$$$$)$8Xl H 6jJ"  pH H 0޽h ? 3380___PPT10.w7 L(  LX L 6ϰjJ"  H@8___PPT9  The trials are identical, and each trial can result in one of the same two possible outcomes, which are denoted by success (S) or failure (F). The trials are independent. The probability of success is constant from trial to trial: denoted by p.h 2~$$$$j$$$8XH L 0޽h ? 3380___PPT10.쿹 P(  P P 6|jJ" P  GBinomial Experiment 2( P 6xjJ" 0p   Suppose each trial of an experiment can result in S or F, but the sampling is without replacement from a population of size N. If the sample size n is at most 5% of the population size, the experiment can be analyzed as though it were exactly a binomial experiment.   22$$$$D$$$$x$l P 6jJ"  `H P 0޽h ? 3380___PPT10.  F>T(  T T 6jJ" : LBinomial Random Variable 2( T 6 jJ" @4 > 2$ ` T 6PjJ"    Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined asP 2*$$ $$/$@ T 6tjJ"  @@D  HX = the number of S s among n trialsh% 2$$$ $$$$l T 6jJ"  P H T 0޽h ? 3380___PPT10.PM;bO XO(  X X 6jJ" pp